Large Deviations for Continuous Time Random Walks

Wang, Wanli; Barkai, Eli; Burov, Stanislav

doi:10.3390/e22060697

Cited by 39 publications

(39 citation statements)

References 64 publications

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“…The first term in the multiplication on the right-hand side of Equation ( 11 ) obviously stems from the equilibrium initial condition under study. We assume that the PDF of the waiting times is analytic for

, thus we can express

as [ 22 , 23 ]

with

an integer number. As an example, consider the case with exponential waiting times, i.e.,

, namely the waiting times at the

states are identically distributed.…”

Section: Resultsmentioning

confidence: 99%

“…Then, the marginal distribution for the displacements follows As we did in Section 2.2.1.1, using the general forms obtained above, i.e., Equation ( 9), Equation (17), Equation (18), Equation (23), and Equation (28), we can analyze P(x, t) in the short and long time limits.…”

Section: Discussionmentioning

confidence: 99%

“…In all the equations above on the right-hand side, we have a multiplication of functions in the Laplace space, this implies convolutions as we transform from s to t. The first term in the multiplication on the right-hand side of Equation ( 11) obviously stems from the equilibrium initial condition under study. We assume that the PDF of the waiting times is analytic for τ → 0, thus we can express ψ ± (τ) as [22,23]…”

Section: Short Time Regimementioning

confidence: 99%

“…Appendix F. A Complementary Deduction of f ± t (T + |1) In this section, we obtain Equation (23) from the definition of conditional probability. The conditional probability of T + , given that N jumps have been made, follows…”

Section: Conflicts Of Interestmentioning

confidence: 99%

“…Phenomenological approaches are diffusing diffusivity models, in which non-Gaussianity is obtained by coupled stochastic differential equations with random diffusion coefficients [ 7 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ], and path integrals formalism for Brownian motion in the presence of a sink [ 21 ]. More recently, theoretical frameworks describing this behavior emerged from continuous time random walk (CTRW) approaches employing large deviations theory [ 22 , 23 , 24 ] and microscopical models like molecular dynamics of tracer particles in polymer networks [ 25 , 26 ] and interacting particles with fluctuating sizes [ 27 , 28 , 29 ], the so-called Hitchhiker model [ 28 ].…”

Section: Introductionmentioning

confidence: 99%

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Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Hidalgo-Soria

Barkai

Burov

2021

Entropy

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We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants, D+>D−, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D−⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

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, thus we can express

as [ 22 , 23 ]

with

an integer number. As an example, consider the case with exponential waiting times, i.e.,

, namely the waiting times at the

states are identically distributed.…”

Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Short Time Regimementioning

confidence: 99%

Section: Conflicts Of Interestmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

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2021

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Heat Kernel Asymptotics for Scaling Limits of Isoradial Graphs

Schwarz,

Sturm,

Wardetzky

2024

Potential Anal

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We consider the asymptotics of the discrete heat kernel on isoradial graphs for the case where the time and the edge lengths tend to zero simultaneously. Depending on the asymptotic ratio between time and edge lengths, we show that two different regimes arise: (i) a Gaussian regime and (ii) a Poissonian regime, which resemble the short-time asymptotics of the heat kernel on (i) Euclidean spaces and (ii) graphs, respectively.

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Molecular diffusion in ternary poly(vinyl alcohol) solutions

Majerczak

Squillace

Shi

et al. 2021

Front. Chem. Sci. Eng.

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The diffusion kinetics of a molecular probe—rhodamine B—in ternary aqueous solutions containing poly(vinyl alcohol), glycerol, and surfactants was investigated using fluorescence correlation spectroscopy and dynamic light scattering. We show that the diffusion characteristics of rhodamine B in such complex systems is determined by a synergistic effect of molecular crowding and intermolecular interactions between chemical species. The presence of glycerol has no noticeable impact on rhodamine B diffusion at low concentration, but significantly slows down the diffusion of rhodamine B above 3.9% (w/v) due to a dominating steric inhibition effect. Furthermore, introducing surfactants (cationic/nonionic/anionic) to the system results in a decreased diffusion coefficient of the molecular probe. In solutions containing nonionic surfactant, this can be explained by an increased crowding effect. For ternary poly(vinyl alcohol) solutions containing cationic or anionic surfactant, surfactant—polymer and surfactant—rhodamine B interactions alongside the crowding effect of the molecules slow down the overall diffusivity of rhodamine B. The results advance our insight of molecular migration in a broad range of industrial complex formulations that incorporate multiple compounds, and highlight the importance of selecting the appropriate additives and surfactants in formulated products.

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Large Deviations for Continuous Time Random Walks

Cited by 39 publications

References 64 publications

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Heat Kernel Asymptotics for Scaling Limits of Isoradial Graphs

Molecular diffusion in ternary poly(vinyl alcohol) solutions

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